Further increasing \(Ra\)  leads to several modes interacting and eventually to a turbulent state. Figure REF  shows an example of turbulent flows at \(E=10^{-6}\) , \(=0.001\)  and \(Ra=5.0\times 10^7\) .  The Rossby number in this case is calculated to be \(Ro=2.9\times 10^{-2}\) , suggesting the turbulence convection is strongly influenced by the rotation. The axial vorticity (figure REF (a)) exhibits both small scales and large scale structures elongated along the rotation axis. These large scale vortices located near the equator would merge into a single vortex at the center when the stress-free boundary condition is used [1]}. Figure REF (b) shows time-evolution of the axial velocity along two dashed lines in figure REF (a), which exhibits wave-like behaviors with typical frequencies within the spectrum of inertial waves (see also figure REF (c)). Figure REF (c) shows the energy spectrum as a function of the azimuthal wavenumber \(m\) .
We can see from the spectrum that there is a dominant toroidal \(m=0\)  component, which represents a strong mean zonal flow.
For smaller scales (\(20\lesssim m \lesssim \)  200), we see that the energy spectrum follows a power law of \(m^{-2}\) , which is reminiscent of the energy spectrum of inertial wave turbulence driven by elliptical instability [2]}. The power law \(m^{-2}\)  is also predicted by [3]} using a phenomenological approach for rotating turbulence. These observations suggest that the interaction of inertial waves plays an important role even in the turbulent regime, though further investigations are required in future.
